# Matrices Chapter 3 Class 12 Maths NCERT Textbook PDF

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### Chapter 3: Matrices

#### 3.1 Introduction

The knowledge of matrices is necessary for various branches of mathematics. Matrices are one of the most powerful tools in mathematics.

This mathematical tool simplifies our work to a great extent when compared with other straightforward methods.

The evolution of the concept of matrices is the result of an attempt to obtain compact and simple methods of solving a system of linear equations. Matrices are not only used as a representation of the coefficients in a system of linear equations, but the utility of matrices far exceeds that use.

Matrix notation and operations are used in electronic spreadsheet programs for personal computers, which in turn are used in different areas of business and science like budgeting, sales projection, cost estimation, analyzing the results of an experiment, etc.

Also, many physical operations such as magnification, rotation, and reflection through a plane can be represented mathematically by matrices.

Matrices are also used in cryptography. This mathematical tool is not only used in certain branches
of sciences, but also in genetics, economics, sociology, modern psychology, and industrial management.

In this chapter, we shall find it interesting to become acquainted with the fundamentals of matrix and matrix algebra.

#### 3.2 Matrix

Suppose we wish to express the information that Radha has 15 notebooks. We may express it (15) with the understanding that the number inside is the number of notebooks that Radha has.

Now, if we have to express that Radha has 15 notebooks and 6 pens.

We may express it as [15 6] with the understanding that the first number inside [ ] is the number of notebooks while the other one is the number of pens possessed by Radha.

### NCERT Solutions Class 12 Maths Chapter 3 Matrices

Question 1:

In the matrix , write:

(i) The order of the matrix (ii) The number of elements,

(iii) Write the elements a13a21a33a24a23

(i) In the given matrix, the number of rows is 3 and the number of columns is 4. Therefore, the order of the matrix is 3 × 4.

(ii) Since the order of the matrix is 3 × 4, there are 3 × 4 = 12 elements in it.

(iii) a13 = 19, a21 = 35, a33 = −5, a24 = 12, a23 = Question 2:

If a matrix has 24 elements, what is the possible order it can have? What, if it has 13 elements?

We know that if a matrix is of the order m × n, it has mn elements. Thus, to find all the possible orders of a matrix having 24 elements, we have to find all the ordered pairs of natural numbers whose product is 24.

The ordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and
(6, 4)

Hence, the possible orders of a matrix having 24 elements are:

1 × 24, 24 × 1, 2 × 12, 12 × 2, 3 × 8, 8 × 3, 4 × 6, and 6 × 4

(1, 13) and (13, 1) are the ordered pairs of natural numbers whose product is 13.

Hence, the possible orders of a matrix having 13 elements are 1 × 13 and 13 × 1.

Question 3:

If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

We know that if a matrix is of the order m × n, it has mn elements. Thus, to find all the possible orders of a matrix having 18 elements, we have to find all the ordered pairs of natural numbers whose product is 18.

The ordered pairs are: (1, 18), (18, 1), (2, 9), (9, 2), (3, 6,), and (6, 3)

Hence, the possible orders of a matrix having 18 elements are:

1 × 18, 18 × 1, 2 × 9, 9 × 2, 3 × 6, and 6 × 3

(1, 5) and (5, 1) are the ordered pairs of natural numbers whose product is 5.

Hence, the possible orders of a matrix having 5 elements are 1 × 5 and 5 × 1.

Question 4:

Construct a 2 × 2 matrix, , whose elements are given by:

(i) (ii) (iii) In general, a 2 × 2 matrix is given by (i) Therefore, the required matrix is (ii) Therefore, the required matrix is (iii) Therefore, the required matrix is Question 5:

Construct 3 × 4 matrices, whose elements are given by

(i) (ii) In general, a 3 × 4 matrix is given by (i) Therefore, the required matrix is (ii) Therefore, the required matrix is 